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MATH 1100: QUANTITATIVE ANALYSIS TEST #3 (VERSION A) 1. Suppose that the demand for a product is given by pq + p = 4. (a) Find the elasticity when p = $2. (b) What type of elasticity is this (unitary, elastic, inelastic)? (c) How would revenue be affected by an increase in price? Solution: (a) Solving for q in terms of p: q= 4−p . p To find the elasticity p dq q dp 4 = . 4−p η=− At p = 2 this gives η = 2. (b) The demand is elastic (η > 1). (c) When the demand is elastic, revenue decreases with price increase. 2. Total revenue is R(x) = 18x − x2 where x is the number of units sold. (a) What number of units sold maximizes revenue? (b) What is the maximum revenue? Solution: (a) x = 9 (right in between the two zeros of R(x), and you don’t even need calculus to see that). (b) R(x) = R(9) = 81. 3. (a) Find critical points, values at the critical points, local maxima, local minima and inflection points of 3 C(x) = x3 − x2 − 18x + 6 2 defined for −4 ≤ x ≤ 4. (b) Graph C(x) for −4 ≤ x ≤ 4. Solution: (a) C 0 (x) = 3x2 − 3x − 18 = 3(x + 2)(x − 3) Date: October 18, 2001. 1 2 MATH 1100: QUANTITATIVE ANALYSIS TEST #3 (VERSION A) 20 10 –4 –3 –2 –1 1 x 2 3 4 0 –10 –20 –30 so the critical points are x = −4, −2, 3, 4. The critical values are C(−4) = −10 C(−2) = 28 69 1 C(3) = − = −34 − 2 2 C(4) = −26. So x = −4 is a local min, x = −2 is a global max, x = 3 a global min and x = 4 is a local max. 1 00 C (x) = 6x − 3 = 6 x − . 2 So the inflection is at x = 1/2. (b) See figure 3b. 4. Suppose that the number x of units produced per month depends on the number y of thousands of dollars invested with x = 3y + 2y 2 . At what rate will production increase if $10000 is invested and the the investment capital is increasing at a rate of $10 per month? Solution: dx dx dy = dt dy dt dy = (3 + 4y) dt MATH 1100: QUANTITATIVE ANALYSIS TEST #3 (VERSION A) 3 Plug in y = 10 (thousand dollars) and dy/dt = 1/1000 (thousands of dollars per month): dx 1 = (3 + 4(10)) dt 1000 43 = . 1000 5. Suppose x + exy = y 2 . Find where x = 0 and y = 1. Solution: dy dy 1 + exy y + x = 2y . dx dx Plug in x = 0 and y = 1, dy = 1. dx dy dx